Mathematics Area - SISSA - Quantum Groups and Noncommutative Geometry

Noncommutative Geometry 1

Daniele Agostinelli — Mon, 24 Jun 2019 16:28:52 +0000

Course Type:

PhD Course

Academic Year:

2019-2020

Period:

October - January

Duration:

30 h

Additional Material:

NCG_I.pdf

Research Group:

Geometry and Mathematical Physics

Location:

A-136

Next Lectures:

KK-theory in geometry and physics

Paolo Antonini — Sat, 28 Apr 2018 13:32:51 +0000

Research Group:

Geometry and Mathematical Physics

Bram Mesland

Institution:

Universität Bonn

Schedule:

Friday, May 4, 2018 - 11:00 to 12:00

Abstract:

Kasparov's $KK$-theory is a bivariant homology theory for $C^*$-algebras providing a general framework for index theory. Cycles for the theory come from the abstract analogue of a first-order differential operator. After an exposition of the theory for non-specialists, I will discuss some recent results. These include analytic properties of self-adjoint operators, applications of $KK$-theory in the theory of arithmetic manifolds and automorphic forms and the physics of topological insulators.

Translation invariant Dirichlet forms in the context of locally compact quantum groups

Paolo Antonini — Tue, 17 Apr 2018 06:27:37 +0000

Research Group:

Geometry and Mathematical Physics

Adam Skalski

Institution:

Impan (Warsaw)

Schedule:

Monday, April 23, 2018 - 14:15 to 15:15

Abstract:

Since the work of Cipriani on one hand and Goldstein and Lindsay on the other in the 1990s it is known that certain natural class of symmetric Markov semigroups on a von Neumann algebra $M$ equipped with a faithful normal state admits extensions to associated Haagerup $L^p$-spaces and is characterised via a Dirichlet property of the generating quadratic form on the $L^2$-space. Recently Cipriani, Franz and Kula studied a special class of such semigroups associated to compact quantum groups. In this talk I will discuss how their results extend to the framework of locally compact quantum groups, where two new important technical features appear: there is no natural "algebraic" domain for the generator and one needs to work with weights, as opposed to finite states (using the extension of the appropriate Dirichlet form result provided by Goldstein and Lindsay). I will also present some applications of Dirichlet forms to the study of geometric properties of quantum groups. This is based on joint work with A.Viselter.

Line Bundles over Multipullback Quantum Complex Projective Spaces

Paolo Antonini — Sat, 31 Mar 2018 07:01:59 +0000

Research Group:

Geometry and Mathematical Physics

Albert Jeu-Liang Sheu

Institution:

University of Kansas

Schedule:

Wednesday, April 11, 2018 - 11:30 to 12:30

Abstract:

Among the popular and well analyzed quantum spaces $X_{q}$ with their $K$-groups $K^{i}\left( X_{q}\right) \equiv K_{i}\left( C\left(X_{q}\right) \right) $ already computed, are the multipullback quantum odd-dimensional spheres $\mathbb{S}_{H}^{2n+1}$ and the associated quantum complex projective spaces $\mathbb{P}^{n}\left( \mathcal{T}\right) $, introduced and studied by Hajac, Kaygun, Nest, Pask, Sims, and Zielinski. In noncommutative geometry, finitely generated projective modules (f.g.p.m.) over $C\left( X_{q}\right)$, efficiently encoded by projections over $C\left( X_{q}\right) $, are viewed as (quantum) vector bundles over $X_{q}%$, and are classified up to stable isomorphism by the positive cone of $K_{0}\left( C\left( X_{q}\right) \right) $. With the $K_{0}$-groups of $C\left( \mathbb{S}_{H}^{2n+1}\right) $ and $C\left( \mathbb{P}^{n}\left(\mathcal{T}\right) \right) $ known, it is natural to seek the classification of vector bundles over $\mathbb{S}_{H}^{2n+1}$ and $\mathbb{P}^{n}\left(\mathcal{T}\right) $ up to isomorphism. Following an approach popularized by Curto, Muhly, and Renault, we first realize $C\left( \mathbb{P}^{n}\left( \mathcal{T}\right) \right) $ and $C\left( \mathbb{S}_{H}^{2n+1}\right) $ as groupoid $C^*$-algebras to better understand their structures in the framework of groupoid $C^*$-algebras, which also provides a convenient context for discussing projections over them. Then we apply Rieffel's theory of stable ranks to derive some answers regarding the classification of f.g.p.m. over the $n$-dimensional Toeplitz algebra $\mathcal{T}^{\otimes n}$, $C\left( \mathbb{S}_{H}^{2n+1}\right) $, and $C\left( \mathbb{P}^{n}\left( \mathcal{T}\right) \right) $. In particular, as concrete direct sums of elementary projections over $C\left(\mathbb{P}^{n}\left( \mathcal{T}\right) \right) $, we can identify those distinguished quantum line bundles $L_{k}$ over $\mathbb{P}^{n}\left(\mathcal{T}\right) $ for $k\in\mathbb{Z}$ that were constructed from quantum principal $U\left( 1\right) $-bundles by Hajac and his collaborators,rendering their module structures transparent.

The Kasparov product on open manifolds

Federico Pichi — Thu, 15 Mar 2018 10:49:06 +0000

Research Group:

Geometry and Mathematical Physics

Koen van den Dungen

Institution:

SISSA

Schedule:

Monday, March 19, 2018 - 14:00 to 15:30

Abstract:

I will start with a brief introduction to KK-theory and the construction of the Kasparov product. I will then focus my attention on the case of Dirac operators on open manifolds. In particular, I will discuss the complications that arise for the Kasparov product of such operators if we do not assume that the manifolds are complete.

Modular curvature and Hypergeometric functions

Paolo Antonini — Thu, 08 Mar 2018 11:23:29 +0000

Research Group:

Geometry and Mathematical Physics

Yang Liu

Institution:

Max Planck (Bonn)

Schedule:

Monday, April 16, 2018 - 14:00 to 15:00

Abstract:

In noncommutative geometry, an essential question is to extend the notion of metric and curvature in Riemannian geometry to noncommutative spaces in a operator theoretical framework. A fundamental feature, in contrast to Riemannian geometry, is the fact that metrics are parametrized by noncommutative coordinates. In the conformal geometry of noncommutative tori, the new structure in the modular analog of the Gaussian curvature consists of two spectral functions, which compress the ansatz caused by the noncommutativity between the metric coordinate and its derivatives. In the first part of the talk, I will explain the higher dimensional generalization of a fantastic functional equation between them due to Connes and Moscovici. In the second part, I will show that hypergeometric functions are the build blocks of those spectral functions. A surprising discovery, obtained by combining the power of hypergeometric functions and computer algebra systems, is that Connes-Moscovici functional relation can be extended to a continuous family with respect to the dimension parameter.

Callias-type operators in C*-algebras and positive scalar curvature on noncompact manifolds

Paolo Antonini — Tue, 27 Feb 2018 11:44:31 +0000

Research Group:

Geometry and Mathematical Physics

Simone Cecchini

Institution:

Georg-August-Universität, Göttingen

Schedule:

Monday, March 12, 2018 - 14:00 to 15:00

Abstract:

A Dirac-type operator on a complete Riemannian manifold is of Callias-type if its square is a Schrödinger-type operator with a potential uniformly positive outside of a compact set. We discuss the theory of Callias-type operators twisted with Hilbert $C^*$-module bundles and present an index theorem for such operators. As an application, we present an obstruction to the existence of complete Riemannian metrics of positive scalar curvature on noncompact spin manifolds in terms of closed submanifolds of codimension one. In particular, when $N$ is a closed Spin manifold, we show that if the cylinder $N \times \mathbb{R}$ carries a complete metric of positive scalar curvature, then the (complex) Rosenberg index on $N$ must vanish.

Principal fibrations over noncommutative spheres - part 2

Paolo Antonini — Tue, 27 Feb 2018 10:12:47 +0000

Research Group:

Geometry and Mathematical Physics

Xiao Han

Institution:

SISSA

Schedule:

Monday, March 5, 2018 - 14:00 to 15:00

Abstract:

This is the second part of the seminar:

http://www.math.sissa.it/seminar/principal-fibrations-over-noncommutativ...

Abstract:

We present several examples of noncommutative four spheres that are base spaces of SU(2)-principal bundles with noncommutative seven-spheres as total spaces

Differential calculus on Jordan algebras and modules

Paolo Antonini — Tue, 13 Feb 2018 07:14:39 +0000

Research Group:

Geometry and Mathematical Physics

Alessandro Carotenuto

Institution:

SISSA

Schedule:

Monday, February 19, 2018 - 16:00

Abstract:

Jordan algebras and their rapresentations have renewed the attention of the scientific community in the last years. However very few is known about differential calculus on modules of Jordan algebras. In this seminar we shall present some results in this direction, in particular we shall concentrate on some features of differential calculus for modules of $J^8_3.$

Principal fibrations over noncommutative spheres

Paolo Antonini — Sun, 11 Feb 2018 14:23:54 +0000

Research Group:

Geometry and Mathematical Physics

Xiao Han

Institution:

SISSA

Schedule:

Monday, February 26, 2018 - 15:00 to 16:00

Abstract:

We present several examples of noncommutative four spheres that are base spaces of SU(2)-principal bundles with noncommutative seven-spheres as total spaces